Based on the method of separation of variables, a direct approach in the study of soliton perturbations has been developed in our previous paper ͓Phys. Rev. E 54, 6816 ͑1996͔͒. In this paper, we use it to deal with the nonlinear Schrödinger and the sine-Gordon equations under the action of perturbations. Results which differ from those in past papers are obtained. ͓S1063-651X͑98͒00807-1͔ PACS number͑s͒: 42.65.Tg, 42.81.Dpwhere ␣, ␤, x 0 , and 0 are four real parameters which determine the propagating velocity, height ͑as well as width͒, initial position, and initial phase of the soliton, respectively.

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Electronic properties of the generalized Fibonacci lattices are studied using the dynamical system technique, for which the off-diagonal tight-binding model is employed. The matrix and trace maps are obtained and investigated in a unified way. It is found that the energy spectra are Cantor-like and the wavefunctions are critical at many energies. For some systems, it is also shown that there are extended and localized wavefunctions. In addition, according to the degree of spatial extension or localization, two other types of wavefunctions are further distinguished, of which one has the tendency to be extended and the other has the tendency to be localized.

The Burgers equation is one of the most important prototypic models in nonlinear physics. Various exact solutions of the Burgers equation have been found by many methods. However, it is very difficult to find interactive solutions among different types of nonlinear excitations. We develop a generalized tanh function expansion approach, which can be considered as the Bäcklund transformation, to find interactive solutions between the soliton and other types of Burgers waves.

The nonlinear localized vibrational modes in an anharmonic atomic chain with uniform mass but two alternating force constants between nearest-neighbors are stud ied by means of multiple-scale expansion. This atomic chain models the vibration s of an arrow of atoms in the direction of a diamond-structure lattice or a molecular chain. It is shown that the distribution of the atomic displacements is governed by a perturbed nonlinear Schrdinger equation, and both the statio nary and moving solutions are obtained. The results are somewhat different from that of the diatomic chain with uniform force constants but two alternating mass es. The reason may be that the translational symmetry of diamond-structure latti ce is comparetively lower.

Supplemental information associated with this articlecan be found [ 1 3 0 _ T D $ D I F F ] online at https://doi.

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